Application of fuzzy analytic hierarchy process model on determination of optimized pile-type

Abstract

Pile-type selection is a very important stage of foundation design, and there are many field factors influencing the decision of pile-type selection. Since there is a limitation of traditional “major factors method” to satisfy the requirement of modern foundation construction, this study presents an efficient approach, in which analytic hierarchy process (AHP) is employed. AHP is a multiple criteria decision-making tool that has been applied in many fields related to the decision-making, e.g., in the field of economics, marketing, sociology, etc. However, it is rarely reported that AHP is applied in the field of civil engineering for decision making. In this study, AHP combined with fuzzy synthetic evaluation method is employed to select the type of pile used as the foundation of a residential building in Fuzhou, Fujian Province, China. The results show that fuzzy AHP approach is an easy and efficient way for pile-type selection.

Keywords

pile type / fuzzy synthetic evaluation / analytic hierarchy process (AHP) / optimal selection

Introduction

There are widely distributed soft deposits along the coastal region of China. In the coastal region, pile foundation is an easy and efficient way to improve the bearing resistance of the soft subsoil. Selection of pile-type is a very significant stage in the design of piled foundation. There are many factors affecting the determination of pile-type, e.g., building structure, total cost, geological condition, etc. In traditional design, the Chinese engineers usually prefer to use the “major factors method”. With this method, pile-type is selected based on the most significant influential factors, e.g., project cost and construction period. However, the “major factors method”, which does not consider enough influential factors, has limited capability to find the most efficient solution for complex problem. Thus, in order to obtain a reasonable design, it is desirable to investigate the effects of the influential factors on pile-type selection and find a method, which comprises not only all major factors but also its influence degree, respectively.

Meng et al. introduced a multiobjective fuzzy optimization method to analyze the cross-effects in pile-type selection among various influential factors [1]. This approach obtained a better optimal result in pile-type selection; however, the weights in fuzzy synthetic evaluation method are still allocated through subjective judgment. Therefore, when there are many influential factors, it is difficult to allocate the weight correctly. The analytic hierarchy process (AHP) technique developed by Saaty is a multicriteria decision-making method that can be used to allocate scarce resources [2]. There were many of applications of AHP as a decision making tool in various fields, including economics, marketing, sociology, and regional and urban planning, etc. [3-8]. However, very few reported that AHP was used as a decision-making tool in the field of civil engineering, especially in pile foundation design.

In this study, an evaluation approach, which combines fuzzy synthetic evaluation method with AHP, is employed to select the type of pile foundation of a residential building in Fuzhou, Fujian Province, China. Three influential factors, namely, bearing strata, numbers of stories, and environment, are considered; and seven pile-types, namely, precast concrete pile, PHC pipe pile, drilling cast-in-site piles, hand-drilled pile, tube sinking pile, steel tube pile, and H-type steel pile, are analyzed and compared.

Fuzzy AHP approach model

Synthetic evaluation set

Assume that there are q kinds of decision in system, n kinds of decision is composed of decision set D={d₁, d₂,…,d_n}, and D denotes the suggestions of pile-type being evaluated. Optimization makes comparison in decision set D. According to decision set D, m kinds of index form an evaluating index set P={p₁, p₂,…, p_m}, and it denotes evaluation index set of optimizing pile foundation. Every appraisal objects from D to P are expressed by an object eigenvalue matrix BoldItalic [1,9].

(1)\mathit{X}=\left[\begin{array}{llll}{x}_{\mathrm{11}}& x_{\mathrm{12}}& \cdots & x_{\mathrm{1}n}\\ x_{\mathrm{21}}& x_{\mathrm{22}}& \cdots & x_{\mathrm{2}n}\\ ⋮& ⋮& & ⋮\\ x_{m\mathrm{1}}& x_{m\mathrm{2}}& \cdots & x_{mn}\end{array}\right]=(x_{ij}),

where x_ij denotes the eigenvalue from d_j to p_i (i=1, 2,…, m; j=1, 2,…, n).

In order to simplify the problem, only three kinds of evaluation index, namely, bearing strata, numbers of stories and environment, are considered in this study. D is the set of suggestions of pile-type, considering the three aforementioned influential factors.

Relative optimal degree of factors

In the optimizing process, the maximum eigenvalue ∨x_ij and minimum eigenvalue ∧x_ij of the object eigenvalue matrix BoldItalic denote relative value of bound. The decision usually can be divided into two types of eigenvalue. One has the feature that the greater the eigenvalue is the better the type will be, and the other is on the contrary. The relative optimal degree can be calculated by Eqs. (2) and (3):

(2)r_{ij}={\displaystyle \frac{x_{ij}-\underset{j}{\wedge }{x}_{ij}}{\underset{j}{\vee }{x}_{ij}-\underset{j}{\wedge }{x}_{ij}}},

(3)r_{ij}={\displaystyle \frac{\underset{j}{\vee }{x}_{ij}-x_{ij}}{\underset{j}{\vee }{x}_{ij}-\underset{j}{\wedge }{x}_{ij}}},

where r_ij represents relative optimal degree, and ∧ and ∨ represents the bound symbol. Then, the object eigenvalue matrix BoldItalic can be changed into the object relative optimal degree matrix by Eqs. (2) and (3) [9]:

(4)R=\left[\begin{array}{llll}{r}_{\mathrm{11}}& r_{\mathrm{12}}& \cdots & r_{\mathrm{1}n}\\ r_{\mathrm{21}}& r_{\mathrm{22}}& \cdots & r_{\mathrm{2}n}\\ ⋮& ⋮& & ⋮\\ r_{m\mathrm{1}}& r_{m\mathrm{2}}& \cdots & r_{mn}\end{array}\right]=(r_{ij}).

Determination of weight of factors using AHP

The distribution of weights is mainly confirmed by subjective experience in synthetic evaluation. Though AHP is difficult to give precise values, it is a suitable multicriteria decision-making method. It divides the relative influence factors into many layers, e.g., object layer, criteria layer, and index layer, and then, qualitative analysis and quantitative analysis are preceded on these layers [10]. The basic steps of AHP are as follows.

Structural model of AHP

Considering the relationship between various influential factors, the decision problem is divided into several layers, e.g., object layer, criteria layer, and index layer. The system of synthetic evaluation of optimal pile-type is shown in Fig. 1. As shown in Fig. 1, based on this study three most important factors influencing decision in criteria layer are listed, which presented by U₁, U₂, U₃ respectively. Below the criteria layer, it is index layer. With different factors, the index layer is also different. As bearing strata (U₁) in layer 2, it has 4 selections in lower layer. However environment (U₃) only has two.

Judgment matrix

Comparing pairwise the importance of factors under each level, the judgment matrix BoldItalic is conducted by 1 to 9 scale ratings as shown in Table1[2,11].

Single-level sorting and consistency inspection

According to the established judgment matrix, single-level sorting is introduced to deduce the weight of factors under each level for some factors of above level. The weight (AW) is often calculated by the largest latent root method, which is shown in Eq. (5):

(5)\mathrm{A}\mathrm{W}={\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}W,

where W denotes the factors weight of all levels. The largest latent root λ_max is calculated approximately by root-square method. In order to ensure the satisfactory consistency of judgment matrix, the consistency ratio CR of all levels is calculated to making consistency inspection. The consistency inspection is carried out by Eqs. (6) and (7):

(6)\mathrm{C}\mathrm{R}=\mathrm{C}\mathrm{I}/\mathrm{R}\mathrm{I},

(7)\mathrm{C}\mathrm{I}=({\lambda }_{\max }-n)/(n-\mathrm{1}),

where CR is the stochastic consistency rate of the judgment matrix, CI is the general consistency index of the judgment matrix, and RI is the average stochastic consistency rate of the judgment matrix. The values of RI are listed in Table 2 [12,13].

When CR<0.1, the consistency of the judgment matrix is considered as being satisfactory, which means the weight distribution is reasonable. Otherwise, the matrix must be adjusted until the consistency is satisfied.

Total-level sorting and consistency inspection

Total-level sorting is introduced to determine the sequence weight of factors under each level for object layer. The sorting process is carried out from the highest level to the lowest level layer by layer. Therefore, the consistency inspection of total-level sorting is also processed layer by layer.

Evaluation of fuzzy AHP

After determining the object relative optimal degree matrix R and the weigh index W, the relative optimal degree model of decision, shown in Eq. (6), is deduced to Eq. (8) by optimization criteria [9]:

(8)u_j={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\left\{{\displaystyle \frac{{{\displaystyle \sum _{i=\mathrm{1}}^m[w_i(\mathrm{1}-r_{ij})]}}^p}{{{\displaystyle \sum _{i=\mathrm{1}}^m(w_i{r}_{ij})}}^p}}\right\}}^{2/p}}},

where u_j denotes the relative optimal degree of decision j for priority, and p denotes the distance parameter, in this study p = 1 (Hamming distance). The analytical process is incorporated in a computer software based on MS Visual Basic. All aforementioned procedures are incorporated into the software developed by Zhang et al. [14,15]

Analysis of case history

According to the engineering data of pile foundation in Fuzhou area, the database analysis platform of pile foundation in this area is established. In this platform, the aforementioned three influential factors of pile selection for Fuzhou’s actual situation are considered.

Project description

The project discussed in this study is a residential building with 20-storey in Fujian Province. The hand-drilled pile is adopted as the pile foundation. Geological condition of this project is shown in Table 3.

Analysis of bearing strata

According to the geological condition, the expert system of digital pile foundation gives reasonable suggestions on bearing strata. Based on the technical code for building pile foundation and the actual situation in Fuzhou, 45 kinds of bearing strata soil are divided into four categories, stiff clay, dense sand, gravel soil, and soft and weathered rock [16,17]. The suggestions of bearing strata, silt clay, residual sand clay, and intensely weathered granite are analyzed by an expert system, as developed by Zhang.

Selection of pile-type

The reasonable advices on pile-type are determined by the selection of the aforementioned three influential factors. Based on the suggestions of bearing strata and the actual situation of the project, residual sand clay is selected as the support strata. The numbers of stories is chosen from 11 to 30, and the environment influences are the vibration and noise. According to the actual condition, the types of pile foundations are divided into seven categories in this study, precast concrete pile, PHC pipe pile, punching pouring piles (drilling cast-in-site piles), hand-drilled pile, tube sinking pile, steel tube pile, and H-type steel pile. With the process proposed in the aforementioned part, six types of pile foundation, precast concrete pile, PHC pipe pile, punching pouring piles, hand-drilled pile, tube sinking pile, and steel tube pile have been obtained last, as shown in Fig. 2.

Establishment of structural model of layers in pile-type optimization

In this project, the weight values of relative influential factors are determined by AHP method. The structural model of all selected factors is shown in Fig. 2.

Determination of weight index of various influential factors

According to the basic evaluation step of AHP and the structural model of layers, the judgment matrixes of all layers are built in Tables 4 to 7. All of the matrixes are based on the consulted suggestion from expert system [15].

With the calculated results listed in Tables 4 to 7, the weight of total-level sorting of C-A is W = (0.051, 0.081, 0.199, 0.319, 0.026, 0.078, 0.175, 0.051, 0.038). The consistency check is shown as

(9)

The consistency of the judgment matrix is satisfactory, where b_j denotes the weight of U_i.

Optimization of pile-type

According to the selected influential factors and suggestions of pile-type, the optimization of pile-type is based on fuzzy AHP approach. In Table 1, the optimized method is the secondary fuzzy process. However, the index in the optional index layer of every criteria layer should only be selected once for each analysis, so that the relationship between the criteria layer and the index layer shown in Fig. 2 becomes one-to-one correspondence, and the optimized method still belongs to single level fuzzy evaluation. Consequently, hand-drilled pile has been chosen as the most efficient pile foundation used in this project.

At last, the hand-drilled pile is selected by analysis in this project. It is agrees well with the engineering field, so the proposed approach is a useful tool in pile-type selection.

Conclusions

This study presented a case history of pile-type selection in foundation design in Fuzhou, Fujian Province, China. Due to the limitation of traditional “major factors method”, AHP approach has been introduced in this study. Based on the mathematic analysis and field investigation, the following conclusion can be drawn:

1) AHP, as a tool at the hands of decision makers and researchers, is one of the most widely used multiple criteria decision-making approaches. Based on the analytical results in this study, the fuzzy AHP approach has been proved to be an easy and efficient tool for pile-type selection.

2) According to analyzing process, the optimal result is determined by the selection of judgment matrix. For construction requirement, seven types of pile foundation, namely, concrete precast pile, PHC pipe pile, drilling cast-in-site piles, hand-drilled pile, tube sinking pile, steel tube pile, and H-type steel pile, and three influential factors, namely, bearing strata, numbers of stories, and environment, are considered in this judgment matrix. The types of pile foundation and influential factors, selected based on the analyses, are coincident with the real construction project in Fuzhou, which ensures the precision of analysis results.

3) In this study, only three influential factors of pile-type selection are considered. Though the weights are determined by AHP, the optimal process still belongs to single level fuzzy evaluation. As a matter of fact, the numbers of influential factors should be considered in the actual selection pile-type, and the relationship between each other is not a one-to-one correspondence. In the further study, more integrated database of influential factors should be established for multilevel fuzzy synthetic evaluation analysis.

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